  We explain the origin of bound states in the continuum (BICs) in a planar
grating waveguide, in particular, a mechanism for formation of degenerate BICs,
via the analytical theory of the infinite-grating eigenmodes. Conventional
symmetry-protected BICs are formed at normal incidence mainly by a single
infinite-grating eigenmode that has an odd spatial parity on both sides of the
BIC resonance. The odd parity is the reason for a cutoff from the
radiation-loss channel and appearance of such BICs. The mechanism of emergence
of a degenerate BIC in a vicinity of a degenerate frequency of two
infinite-grating eigenmodes is different. The degenerate BIC is formed by an
anti-phased coherent superposition of two crossing infinite-grating eigenmodes
both of which possess a mixed parity and experience parity symmetry breaking as
the frequency scans through the degeneracy point. In this case a cutoff from
the radiation-loss channel and extremely high-Q narrow resonance is achieved
due to the destructive interference of the two crossing eigenmodes.
Implementation of such a mechanism can be instructive for designing BICs in
other photonic crystals and structures.
