The expressions of the dispersion relationship and space localization in the one-dimensional photonic crystal (1DPC) with symmetric semi-infinite period number and a central defect layer are deduced, which are valid for the defect 1DPC consisting of materials with arbitrary frequency-dependence. Based upon them, we found that the eigenfrequency of defect state is red-shifted with the increasing geometry thickness of defect layer in right-handed materials. On the contrary, the frequency is blue-shifted when the defect layer is a left-handed material. Moreover, the defect state is more localized for symmetric semi-infinite period structures consisting of metamaterials than those of conventional right-handed materials.
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