Wannier functions of defective photonic crystals with broken translational symmetry

Wannier functions offer a complete and well-suited real-space description for perfect crystals, based on localized states. With broken crystalline translational symmetry, however, direct unitary transformation from Bloch functions is not straightforward. Here we present an explicit construction of localized defect states and Wannier functions for defective photonic crystals in the absence of translational symmetry. At first, using functions derived from the corresponding perfect crystals with inversion symmetry, we construct localized defect states at the defect frequencies in the first photonic bandgap of three defective crystals. Then we show that it is possible to construct the exponentially localized Wannier functions for the two defect configurations lacking translational symmetry. These findings are validated through comparisons with the real-space magnetic fields obtained from a commercial electromagnetic simulator. Even though very similar localization characteristics from both fields and Wannier functions are observed for low photonic band indices, they begin to show deviations with increasing band index. We discuss the efficacy of the photonic Wannier functions, accurately representing the exponentially localized mode formed at the defect sites.