Degeneracy analysis for a supercell of a photonic crystal and its application to the creation of band gaps.

A method is introduced to analyze the degeneracy properties of the band structure of a photonic crystal by making use of supercells. The band structure associated with a supercell of a photonic crystal has degeneracies at the edge of the Brillouin zone if the photonic crystal has some kind of point group symmetry. The E-polarization and H-polarization cases have the same degeneracies for a two-dimensional (2D) photonic crystal. Two theorems on degeneracies in the band structure associated with the supercell are given and proved. These degeneracies can be lifted to create photonic band gaps by changing the translation group symmetry of the photonic crystal (the point group symmetry of the photonic crystal may remain unchanged), which consequently changes the transform matrix between the supercell and the smallest unit cell. The existence of photonic band gaps for many known 2D photonic crystals is explained through the degeneracy analysis. Some structures with large band gaps are also found through the present degeneracy analysis.