Bloch state scattering in the Brillouin zone

Systems with space-periodic Hamiltonians have unique scattering properties, just as time-periodic ones do. We use Wigner-Eisenbud (reaction matrix) scattering theory to consider a two-dimensional scattering system in which one dimension is a periodic lattice and the other is localized in space. The scattering and decay properties can then be described by sets of channels, where sets are indexed by Bloch momenta and channels within sets are indexed by incident waves' quantized kinetic energy parallel to the lattice. In the case where the lattice unit cell has reflection symmetry we find that the lattice can sustain formation of antisymmetric bound states in the continuum. Breaking the unit cell symmetry then causes those bound states to become quasibound and slowly decay. We also find in the lowest energy channel that reflection probability increases when varying Bloch momentum.