Light trapping above the light cone in one-dimensional array of dielectric spheres

We demonstrate bound states in the first TE and TM diffraction continua (BSC) in a linear periodic array of dielectric spheres in air above the light cone. We classify the BSCs according to the symmetry specified by the azimuthal number $m$, the Bloch wave vector $\beta$ directed along the array, and polarization. The most simple symmetry protected TE and TM polarized BSCs have $m=0$ and $\beta=0$ and occur in a wide range of the radius of the spheres and dielectric constant. More complicated BSCs with $m\neq 0$ and $\beta=0$ exist only for a selected radius of spheres at a fixed dielectric constant. We also find robust Bloch BSCs with $\beta\neq 0, m=0$. We present also the BSCs embedded into two and three diffraction continua. We show that the BSCs can be easily detected by the collapse of Fano resonance for scattering of electromagnetic plane waves by the array.