We systematically study a collection of refractive phenomena that can
possibly occur at the interface of a two-dimensional photonic crystal, with the
use of the wave vector diagram formalism. Cases with a single propagating beam
(in the positive or the negative direction) as well as cases with birefringence
were observed. We examine carefully the conditions to obtain a single
propagating beam inside the photonic crystal lattice. Our results indicate,
that the presence of multiple reflected beams in the medium of incidence is
neither a prerequisite nor does it imply multiple refracted beams. We
characterize our results in respect to the origin of the propagating beam and
the nature of propagation (left-handed or not). We identified four distinct
cases that lead to a negatively refracted beam. Under these findings, the
definition of phase velocity in a periodic medium is revisited and its physical
interpretation discussed. To determine the ``rightness'' of propagation, we
propose a wedge-type experiment. We discuss the intricate details for an
appropriate wedge design for different types of cases in triangular and square
structures. We extend our theoretical analysis, and examine our conclusions as
one moves from the limit of photonic crystals with high index contrast between
the constituent dielectrics to photonic crystals with low modulation of the
refractive index. Finally, we examine the ``rightness'' of propagation in the
one-dimensional multilayer medium, and obtain conditions that are different
from those of two-dimensional systems.Comment: 65 pages, 17 figures, submitted to Phys. Rev.
