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For the waveguide having a pillar-missing-line defect in a photonic crystal (PhC) with dielectric pillars in air, a pillar-missing-line defect can support the low-loss transmission when the mode frequency falls inside the photonic bandgap (PBG). In this paper, the waveguide with the single-missing-hole-line defect in the silicon on insulator (SOI) PhC slab, which has square air holes in silicon on a square array, is studied by the FDTD simulation. It seems that even in this structure, a high transmission coefficient is obtained in the bandgap-guided mode. However, a low-loss transmission was observed at outside the PBG. This is the propagation of the index-guided mode based on the refractive index of the core being larger than average index of the cladding medium. Appropriately, choosing the design parameters (the silicon core is 1000-nm thick, lattice constant is 500-nm, and air hole size is 250-nm square), we find that the present PhC has a TE gap ranging from 95-THz to 135-THz. These values approximately agree with the 2D theoretical gap ranging from 92-THz to 133-THz because the silicon layer is thick. The low-loss transmissions are obtained at the frequencies outside the PBG in the longitudinal and transverse directions except for the low-loss transmission frequency in the oblique direction on the original PhC. This is the propagation of the index-guided mode. Since the omni-directional bandgap of PhC with a square lattice is generally smaller than that for a triangular lattice, it is considered that PhC with the square lattice is unsuitable to forming sharp waveguide bends. However, in this paper, we propose the fundamental structure of a ring resonator with sharp U-turns such that the possibility of obtaining a high Q value can also be expected. Assuming a 300-nm-thick silicon layer, the 420-nm lattice constant and 300-nm square air holes, we find good optical confinement near 210-THz. At the resonance of 211-THz, we obtain the Q-value of 100 from the saturated tuning curve in the limit to the Fourier transform. These facts indicate that the resonator with sharp U-turns has Q value greater than 100. Since the leakage to the lower and upper layers is not considered in the present study, the possibility of obtaining a high Q value can also be expected.
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