Abstract
We compare the extra modes introduced by plane-wave expansion with the two-dimensional (2D) and the three-dimensional (3D) supercell approximation method, and analyze the differences between those two kinds of extra modes. In the band structure of 2D photonic crystal (PhC) using the 2D supercell approximation, the extra modes are created through band folding, and those extra modes can really exist in the 2D PhC. However, in the band structure of a 2D PhC slab using the 3D supercell approximation, the extra modes are mainly generated by the coupling between supercells, and cannot exist in the real 2D PhC slab, thus these extra modes are defined as fake modes in our paper. Through a detailed analysis of the origin of the fake modes and of the energy distribution of the fake modes and of the real modes, we propose an improved fake mode free plane-wave expansion and three planar slab waveguides method which removes the fake modes. This is the first time such a fake mode free band structure is presented. This result can be definitely helpful in designing 3D devices.
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