  Guided modes of an open periodic waveguide, with a periodicity in the main
propagation direction, are Bloch modes confined around the waveguide core with
no radiation loss in the transverse directions. Some guided modes can have a
complex propagation constant, i.e. a complex Bloch wavenumber, even when the
periodic waveguide is lossless (no absorption loss). These so-called complex
modes are physical solutions that can be excited by incident waves whenever the
waveguide has discontinuities and defects. We show that the complex modes in an
open dielectric periodic waveguide form bands, and the endpoints of the bands
can be classified to a small number of cases, including extrema on dispersion
curves of the regular guided modes, bound states in the continuum, degenerate
complex modes, and special diffraction solutions with blazing properties. Our
study provides an improved theoretical understanding on periodic waveguides and
a useful guidance to their practical applications.
