Interleaving two-dimensional lattices to create three-dimensional photonic bandgap structures

A two-dimensional lattice consists of an inhomogeneous dielectric medium which has periodic variations of permittivity along two linear directions, and is uniform in the third dimension. The paper considers a class of three-dimensional photonic bandgap materials formed by the interleaving of a pair of two-dimensional lattices whose uniform directions are mutually orthogonal, the so-called 'woodpile' geometry. It is shown by numerical calculations that the strong polarisation discrimination exhibited by the constituent two-dimensional sublattices in their respective stopbands leads to a completely three-dimensional photonic bandgap crystal in all polarisations when the two sublattices are interleaved. For a general class of such synthetic crystals having arbitrary two-dimensional distributions in the sublattices, many properties of the three-dimensional interleaved crystal can be predicted directly from a knowledge of the properties of the sublattices, thereby radically the computational cost of their inhomogeneous dielectric medium which periodic variations of permittivity along properties reducing me determination.