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Photonic Crystal (PC) is of an artificially optical material with periodically modulated dielectric, which leads to the generation of photonic band gaps (PBGs). Furthermore, one-dimensional (1-D) PCs with structural defects have received extensive interest because they may serve as natural waveguide and only allow a narrow range of frequencies of light wave to propagate in the PCs. These features should bring about many novel applications to optical devices, such as all-optical circuit, dielectric mirrors, Fabry-Perot filters and distributed feedback lasers, etc. Recently, Fink et al. reported that the 1-D PC possesses the feature of omnidirectional reflection for all the polarizations of incident light, which provides ability to reflect the incident light with arbitrary incident angle. Therefore, this novel property furnishes a wide platform for practical applications. In this presentation, we focus on the investigation of the dispersion spectra in the 1-D PCs with structural defect layer made by the left-handed material with the use of transfer-matrix method. We study the variations of the dispersion spectra of photons when changing the thickness and the refractive index (negative or positive) of the defect layer. It is found that the defect modes can be engineered by inserting the defect layer into the PCs and changing the thickness and the refractive index of defect layer, for instance, the defect layer made by the left-handed material or right-handed material.