Photonic band structure calculations using scattering matrices - art. no. 046603

We consider band structure calculations of two-dimensional photonic crystal s treated as stacks of one-dimensional gratings. The gratings are character ized by their plane wave scattering matrices, the calculation of which is w ell established. These matrices are then used in combination with Bloch's t heorem to determine the band structure of a photonic crystal from the solut ion of an eigenvalue problem. Computationally beneficial simplifications of the eigenproblem for symmetric lattices are derived, the structure of eige nvalue spectrum is classified, and, at long wavelengths, simple expressions for the positions of the band gaps are deduced. Closed form expressions fo r the reflection and transmission scattering matrices of finite stacks of g ratings are established. A new, fundamental quantity, the reflection scatte ring matrix, in the limit in which the stack fills a half space, is derived and is used to deduce the effective dielectric constant of the crystal in the long wavelength limit.