  Using an extended plane-wave-based transfer-matrix method, the photonic band
structures and the corresponding transmission spectrum of a two-dimensional
left-handed photonic crystal are calculated. Comparisons between the periodic
structure with a single left-handed cylindric rod are made, and many
interesting similarities are found. It is shown that, due to the localized
surface polaritons presented by an isolated left-handed rod, there exist many
exciting physical phenomena in high-dimensional left-handed photonic crystals.
As direct results of coupling of the localized surface polaritons of
neighboring left-handed rod, a lot of almost dispersionless bands,
anti-crossing behavior, and a zero $\bar{n}$ gap are exhibited in the
left-handed periodic structure. Moreover, in a certain frequency region, except
distorted by a lot of anti-crossing behavior, there exists a continual
dispersion relation, which can be explained by the long-wavelength
approximation. It is also pointed out that high-dimensional left-handed
photonic crystals can be used to design narrow-band filter.
