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 ε₁ to ε₂. For ε₂=1 in air the rod is equivalent to a stack of dielectric disks with permittivity ε₁. Because of rotational and translational symmetries the BICs are classified by orbital angular momentum m and the Bloch wave vector β directed along the rod. For $m=0$ and β=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≠0,β=0 exist only for a selected radius of the rod at a fixed dielectric constant. The existence of robust Bloch BICs with β≠0,m=0 is demonstrated. Asymptotic limits to a homogeneous rod and to very thin disks are also considered.