Based on the method of dispersion characteristics analysis, the Dirac point was found at the center of the Brillion zone in the band structure of the hexagonal lattice photonic crystals, and the phenomenon that both the effective permittivity and permeability approach zero simultaneously at the Dirac point was verified by using the effective medium theory, so that the zero index material can be achieved by the hexagonal lattice photonic crystals at that point. Then, by using the finite element method to perform the numerical simulation, the zero index characteristic presented at or near the Dirac point of the hexagonal lattice photonic crystals was studied. In addition, by designing the concave lens which is made from the hexagonal lattice photonic crystals properly, the results of the sub-wavelength and far-field focusing can be obtained. It is useful to improve the focusing by introducing an antireflection structure to the incident interface of the concave lens, and can improve the effect of focusing and increase the resolution to some extent.