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We consider bound states in the radiation continuum (BSC) above the light cone in an one-dimensional periodic array of dielectric spheres in air. The BSCs are classified by orbital angular momentum m, Bloch wave vector β directed along the array, and polarization. The most simple symmetry protected BSCs have m = 0, β = 0 and occur in a wide range of the radius of spheres and dielectric constant. More sophisticated BSCs with m ̸= 0, β = 0 exist only for a selected radius of the spheres at a fixed dielectric constant. We also show the existence of robust Bloch BSCs with β ̸=0, m = 0. The BSCs with m = 0 can be easily detected by the collapse of Fano resonance in scattering of a plane wave. In response to a plane wave with circular polarization the BSCs with m ̸= 0 give rise to Poynting vector spiralling around the array.
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