  It is well established that a Dirac point of a periodic structure can
bifurcate into in-gap eigenvalues if the periodic structure is perturbed
differently on the two sides of an interface and if a common band gap can be
opened for the two perturbed periodic structures near the Dirac point. This
paper addresses the less-known situation when the perturbation only lifts the
degeneracy of the Dirac point without opening a band gap. Using a
two-dimensional waveguide model, we constructed a wave mode from the
bifurcation of a Dirac point of a periodic waveguide. We proved that when the
constructed mode couples with the Floquet-Bloch modes of quasi-momentum away
from the Dirac point, its associated eigenvalue has a negative imaginary part
and the mode is a resonant mode that can radiate its energy into the bulk. On
the other hand, when the coupling vanishes, the imaginary part of the
eigenvalue turns to zero, and the constructed mode becomes an interface mode
that decays exponentially away from the interface. It is believed that the
developed method can be extended to other settings, thus providing a clear
answer to the problem concerned with the bifurcation of Dirac points.
