A photonic crystal waveguide network can form in a three-dimensional layer-by-layer photonic crystal along three orthogonal directions. We investigate guided-mode band structures for different waveguide configurations by numerical calculations that combine a plane-wave expansion method with a supercell technique. The in-plane waveguide network composed of waveguides along the (100) and (010) directions of the crystal is located in a single layer of the photonic crystal. This structural design is easy to achieve experimentally. One can create these in-plane waveguides either by removing one single rod or by breaking a segment in each of an array of parallel rods. One produces the off-plane waveguide by removing some segments of rods along the (001) stacking direction of the crystal. Single-mode operation of a waveguide can be achieved by appropriate adjustment of geometrical parameters such as the location and the size of the waveguide. Intrinsic lattice symmetries in these photonic crystal waveguides have been fully employed to reduce the complexity of the numerical solution of guided-mode band structures significantly.
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