Three-dimensional analysis of hexagonal structured photonic crystals using oblique coordinates

The band structures of hexagonal photonic crystals with finite height are determined in this paper. Three-dimensional, full vectorial calculations are performed with the Method of Lines as numerical algorithm. To describe the boundaries of the elementary cells correctly, oblique coordinates are introduced. In particular, suitable expressions for the analysis with oblique coordinates and for the three-dimensional case are developed from Maxwell’s equations. A comparison of the numerical results with values from the literature shows the validity of the derived algorithm.