  A semi-Dirac cone refers to a peculiar type of dispersion relation that is
linear along the symmetry line but quadratic in the perpendicular direction.
Here, I demonstrate that a photonic crystal consisting of a square array of
elliptical dielectric cylinders is able to produce this particular dispersion
relation in the Brillouin zone center. A perturbation method is used to
evaluate the linear slope and to affirm that the dispersion relation is a
semi-Dirac type. Effective medium parameters calculated from a boundary
effective medium theory not only explain the unexpected topological transition
in the iso-frequency surfaces occurring at the semi-Dirac point, they also
offer a perspective on the property at that point, where the photonic crystal
behaves as a zero-refractive-index material along the symmetry axis but
functions like at a photonic band edge in the perpendicular direction.
