Abstract:
We consider band structure calculations of two-dimensional photonic crystals treated as stacks of onedimensional
gratings. The gratings are characterized by their plane wave scattering matrices, the calculation of
which is well established. These matrices are then used in combination with Bloch's theorem to determine the
band structure of a photonic crystal from the solution of an eigenvalue problem. Computationally beneficial
simplifications of the eigenproblem for symmetric lattices are derived, the structure of eigenvalue spectrum is
classified, and, at long wavelengths, simple expressions for the positions of the band gaps are deduced. Closed
form expressions for the reflection and transmission scattering matrices of finite stacks of gratings are established.
Anew, fundamental quantity, the reflection scattering 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.
